{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "5a09a7f6",
   "metadata": {},
   "source": [
    "# Exponencial function\n",
    "\n",
    "In this tutoial we show a few Padé approximants of $e^{x}\\,$ using infinite and finite precision."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "id": "28869892",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "from padepy import baker_algorithm as ba\n",
    "from padepy import direct_algorithm as da"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "id": "3c3818e7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e^{x}$"
      ],
      "text/plain": [
       "exp(x)"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var = sp.Symbol(\"x\")\n",
    "func = sp.exp(var)\n",
    "p = 3\n",
    "q = 3\n",
    "func"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "id": "f2fafb2d",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pade [3,3](x) stored at matrix index 6.\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\frac{x^{3}}{120} + \\frac{x^{2}}{10} + \\frac{x}{2} + 1}{- \\frac{x^{3}}{120} + \\frac{x^{2}}{10} - \\frac{x}{2} + 1}$"
      ],
      "text/plain": [
       "(x**3/120 + x**2/10 + x/2 + 1)/(-x**3/120 + x**2/10 - x/2 + 1)"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pade, path = ba.pade(p, q, var, func, 0, False)\n",
    "pade"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "id": "7d43f619",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{x^{6}}{720} + \\frac{x^{5}}{120} + \\frac{x^{4}}{24} + \\frac{x^{3}}{6} + \\frac{x^{2}}{2} + x + 1\\\\\\frac{x^{5}}{120} + \\frac{x^{4}}{24} + \\frac{x^{3}}{6} + \\frac{x^{2}}{2} + x + 1\\\\\\frac{\\frac{x^{5}}{720} + \\frac{x^{4}}{72} + \\frac{x^{3}}{12} + \\frac{x^{2}}{3} + \\frac{5 x}{6} + 1}{1 - \\frac{x}{6}}\\\\\\frac{\\frac{x^{4}}{120} + \\frac{x^{3}}{15} + \\frac{3 x^{2}}{10} + \\frac{4 x}{5} + 1}{1 - \\frac{x}{5}}\\\\\\frac{\\frac{x^{4}}{360} + \\frac{x^{3}}{30} + \\frac{x^{2}}{5} + \\frac{2 x}{3} + 1}{\\frac{x^{2}}{30} - \\frac{x}{3} + 1}\\\\\\frac{\\frac{x^{3}}{60} + \\frac{3 x^{2}}{20} + \\frac{3 x}{5} + 1}{\\frac{x^{2}}{20} - \\frac{2 x}{5} + 1}\\\\\\frac{\\frac{x^{3}}{120} + \\frac{x^{2}}{10} + \\frac{x}{2} + 1}{- \\frac{x^{3}}{120} + \\frac{x^{2}}{10} - \\frac{x}{2} + 1}\\\\\\frac{\\frac{x^{2}}{20} + \\frac{2 x}{5} + 1}{- \\frac{x^{3}}{60} + \\frac{3 x^{2}}{20} - \\frac{3 x}{5} + 1}\\\\\\frac{\\frac{x^{2}}{30} + \\frac{x}{3} + 1}{\\frac{x^{4}}{360} - \\frac{x^{3}}{30} + \\frac{x^{2}}{5} - \\frac{2 x}{3} + 1}\\\\\\frac{\\frac{x}{5} + 1}{\\frac{x^{4}}{120} - \\frac{x^{3}}{15} + \\frac{3 x^{2}}{10} - \\frac{4 x}{5} + 1}\\\\\\frac{\\frac{x}{6} + 1}{- \\frac{x^{5}}{720} + \\frac{x^{4}}{72} - \\frac{x^{3}}{12} + \\frac{x^{2}}{3} - \\frac{5 x}{6} + 1}\\\\\\frac{1}{- \\frac{x^{5}}{120} + \\frac{x^{4}}{24} - \\frac{x^{3}}{6} + \\frac{x^{2}}{2} - x + 1}\\\\\\frac{1}{\\frac{x^{6}}{720} - \\frac{x^{5}}{120} + \\frac{x^{4}}{24} - \\frac{x^{3}}{6} + \\frac{x^{2}}{2} - x + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[       x**6/720 + x**5/120 + x**4/24 + x**3/6 + x**2/2 + x + 1],\n",
       "[                  x**5/120 + x**4/24 + x**3/6 + x**2/2 + x + 1],\n",
       "[ (x**5/720 + x**4/72 + x**3/12 + x**2/3 + 5*x/6 + 1)/(1 - x/6)],\n",
       "[        (x**4/120 + x**3/15 + 3*x**2/10 + 4*x/5 + 1)/(1 - x/5)],\n",
       "[ (x**4/360 + x**3/30 + x**2/5 + 2*x/3 + 1)/(x**2/30 - x/3 + 1)],\n",
       "[       (x**3/60 + 3*x**2/20 + 3*x/5 + 1)/(x**2/20 - 2*x/5 + 1)],\n",
       "[(x**3/120 + x**2/10 + x/2 + 1)/(-x**3/120 + x**2/10 - x/2 + 1)],\n",
       "[      (x**2/20 + 2*x/5 + 1)/(-x**3/60 + 3*x**2/20 - 3*x/5 + 1)],\n",
       "[ (x**2/30 + x/3 + 1)/(x**4/360 - x**3/30 + x**2/5 - 2*x/3 + 1)],\n",
       "[        (x/5 + 1)/(x**4/120 - x**3/15 + 3*x**2/10 - 4*x/5 + 1)],\n",
       "[(x/6 + 1)/(-x**5/720 + x**4/72 - x**3/12 + x**2/3 - 5*x/6 + 1)],\n",
       "[             1/(-x**5/120 + x**4/24 - x**3/6 + x**2/2 - x + 1)],\n",
       "[   1/(x**6/720 - x**5/120 + x**4/24 - x**3/6 + x**2/2 - x + 1)]])"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "path"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f82b93b2",
   "metadata": {},
   "source": [
    "It is possible to input a list with coefficients."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "id": "73f532ac",
   "metadata": {},
   "outputs": [],
   "source": [
    "from padepy import maclaurin as ma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "id": "88ecfbb2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040]"
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = p + q + 1\n",
    "coeffs_list = ma.coefficients(n, var, func)\n",
    "coeffs_list"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "id": "9d97b9ad",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pade [3,3](x) stored at matrix index 6.\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\frac{x^{3}}{120} + \\frac{x^{2}}{10} + \\frac{x}{2} + 1}{- \\frac{x^{3}}{120} + \\frac{x^{2}}{10} - \\frac{x}{2} + 1}$"
      ],
      "text/plain": [
       "(x**3/120 + x**2/10 + x/2 + 1)/(-x**3/120 + x**2/10 - x/2 + 1)"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pade, path = ba.pade(p, q, var, coeffs_list, 0, False)\n",
    "pade"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "831cc33e",
   "metadata": {},
   "source": [
    "If we set $x=1$ we get an approximation of the value $e = 2.7182818284590452353602874713526624977572470936999595749669676277...$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "id": "2406d733",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.71830985915493$"
      ],
      "text/plain": [
       "2.71830985915493"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = 2.7182818284590452353602874713526624977572470936999595749669676277\n",
    "e_approx = pade.subs(var, 1.0)\n",
    "e_approx "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "id": "8e6558bf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.80306958844179 \\cdot 10^{-5}$"
      ],
      "text/plain": [
       "2.80306958844179e-5"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "error = e_approx - e\n",
    "error"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0e6050f8",
   "metadata": {},
   "source": [
    "We can also replace in path variable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "id": "95d0df25",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.71805555555556\\\\2.71666666666667\\\\2.71833333333333\\\\2.71875\\\\2.71825396825397\\\\2.71794871794872\\\\2.71830985915493\\\\2.71875\\\\2.7182320441989\\\\2.71698113207547\\\\2.71844660194175\\\\2.72727272727273\\\\2.71698113207547\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2.71805555555556],\n",
       "[2.71666666666667],\n",
       "[2.71833333333333],\n",
       "[         2.71875],\n",
       "[2.71825396825397],\n",
       "[2.71794871794872],\n",
       "[2.71830985915493],\n",
       "[         2.71875],\n",
       "[ 2.7182320441989],\n",
       "[2.71698113207547],\n",
       "[2.71844660194175],\n",
       "[2.72727272727273],\n",
       "[2.71698113207547]])"
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_approx_path = path.subs(var, 1.0)\n",
    "e_approx_path"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b9000567",
   "metadata": {},
   "source": [
    "It is possible to work with floating precision."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "id": "8b4020a9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Pade [3,3](x) stored at matrix index 6.\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{0.00833333333333333 x^{3} + 0.09999999999999997 x^{2} + 0.4999999999999999 x + 1.0}{- 0.008333333333333339 x^{3} + 0.1 x^{2} - 0.5000000000000001 x + 1.0}$"
      ],
      "text/plain": [
       "(0.00833333333333333*x**3 + 0.09999999999999997*x**2 + 0.4999999999999999*x + 1.0)/(-0.008333333333333339*x**3 + 0.1*x**2 - 0.5000000000000001*x + 1.0)"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precision = 16\n",
    "pade, path = ba.pade(p, q, var, coeffs_list, precision)\n",
    "pade"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "id": "c7f214c1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.001388888888888889 x^{6} + 0.008333333333333333 x^{5} + 0.04166666666666667 x^{4} + 0.1666666666666667 x^{3} + 0.5 x^{2} + 1.0 x + 1.0\\\\0.008333333333333333 x^{5} + 0.04166666666666667 x^{4} + 0.1666666666666667 x^{3} + 0.5 x^{2} + 1.0 x + 1.0\\\\\\frac{0.001388888888888889 x^{5} + 0.01388888888888889 x^{4} + 0.08333333333333333 x^{3} + 0.3333333333333333 x^{2} + 0.8333333333333333 x + 1.0}{1.0 - 0.1666666666666667 x}\\\\\\frac{0.008333333333333334 x^{4} + 0.06666666666666667 x^{3} + 0.3 x^{2} + 0.8 x + 1.0}{1.0 - 0.2 x}\\\\\\frac{0.00277777777777778 x^{4} + 0.03333333333333335 x^{3} + 0.2 x^{2} + 0.6666666666666667 x + 1.0}{0.03333333333333332 x^{2} - 0.3333333333333333 x + 1.0}\\\\\\frac{0.01666666666666667 x^{3} + 0.15 x^{2} + 0.6 x + 1.0}{0.05 x^{2} - 0.4 x + 1.0}\\\\\\frac{0.00833333333333333 x^{3} + 0.09999999999999997 x^{2} + 0.4999999999999999 x + 1.0}{- 0.008333333333333339 x^{3} + 0.1 x^{2} - 0.5000000000000001 x + 1.0}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                                  0.001388888888888889*x**6 + 0.008333333333333333*x**5 + 0.04166666666666667*x**4 + 0.1666666666666667*x**3 + 0.5*x**2 + 1.0*x + 1.0],\n",
       "[                                                              0.008333333333333333*x**5 + 0.04166666666666667*x**4 + 0.1666666666666667*x**3 + 0.5*x**2 + 1.0*x + 1.0],\n",
       "[(0.001388888888888889*x**5 + 0.01388888888888889*x**4 + 0.08333333333333333*x**3 + 0.3333333333333333*x**2 + 0.8333333333333333*x + 1.0)/(1.0 - 0.1666666666666667*x)],\n",
       "[                                                                        (0.008333333333333334*x**4 + 0.06666666666666667*x**3 + 0.3*x**2 + 0.8*x + 1.0)/(1.0 - 0.2*x)],\n",
       "[                (0.00277777777777778*x**4 + 0.03333333333333335*x**3 + 0.2*x**2 + 0.6666666666666667*x + 1.0)/(0.03333333333333332*x**2 - 0.3333333333333333*x + 1.0)],\n",
       "[                                                                                       (0.01666666666666667*x**3 + 0.15*x**2 + 0.6*x + 1.0)/(0.05*x**2 - 0.4*x + 1.0)],\n",
       "[              (0.00833333333333333*x**3 + 0.09999999999999997*x**2 + 0.4999999999999999*x + 1.0)/(-0.008333333333333339*x**3 + 0.1*x**2 - 0.5000000000000001*x + 1.0)]])"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "path"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "id": "c57c09d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.71830985915493$"
      ],
      "text/plain": [
       "2.718309859154930"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_approx = pade.subs(var, 1.0)\n",
    "e_approx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "id": "32a47954",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}2.718055555555556\\\\2.716666666666667\\\\2.718333333333333\\\\2.71875\\\\2.718253968253968\\\\2.717948717948718\\\\2.71830985915493\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[2.718055555555556],\n",
       "[2.716666666666667],\n",
       "[2.718333333333333],\n",
       "[          2.71875],\n",
       "[2.718253968253968],\n",
       "[2.717948717948718],\n",
       "[ 2.71830985915493]])"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e_approx_path = path.subs(var, 1.0)\n",
    "e_approx_path"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84102a8c",
   "metadata": {},
   "source": [
    "We will use bokeh library to plot. Use: pip install bokeh"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "id": "85627fdf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class=\"bk-root\">\n",
       "        <a href=\"https://bokeh.org\" target=\"_blank\" class=\"bk-logo bk-logo-small bk-logo-notebook\"></a>\n",
       "        <span id=\"5376\">Loading BokehJS ...</span>\n",
       "    </div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": "(function(root) {\n  function now() {\n    return new Date();\n  }\n\n  const force = true;\n\n  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n    root._bokeh_onload_callbacks = [];\n    root._bokeh_is_loading = undefined;\n  }\n\nconst JS_MIME_TYPE = 'application/javascript';\n  const HTML_MIME_TYPE = 'text/html';\n  const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n  const CLASS_NAME = 'output_bokeh rendered_html';\n\n  /**\n   * Render data to the DOM node\n   */\n  function render(props, node) {\n    const script = document.createElement(\"script\");\n    node.appendChild(script);\n  }\n\n  /**\n   * Handle when an output is cleared or removed\n   */\n  function handleClearOutput(event, handle) {\n    const cell = handle.cell;\n\n    const id = cell.output_area._bokeh_element_id;\n    const server_id = cell.output_area._bokeh_server_id;\n    // Clean up Bokeh references\n    if (id != null && id in Bokeh.index) {\n      Bokeh.index[id].model.document.clear();\n      delete Bokeh.index[id];\n    }\n\n    if (server_id !== undefined) {\n      // Clean up Bokeh references\n      const cmd_clean = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n      cell.notebook.kernel.execute(cmd_clean, {\n        iopub: {\n          output: function(msg) {\n            const id = msg.content.text.trim();\n            if (id in Bokeh.index) {\n              Bokeh.index[id].model.document.clear();\n              delete Bokeh.index[id];\n            }\n          }\n        }\n      });\n      // Destroy server and session\n      const cmd_destroy = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n      cell.notebook.kernel.execute(cmd_destroy);\n    }\n  }\n\n  /**\n   * Handle when a new output is added\n   */\n  function handleAddOutput(event, handle) {\n    const output_area = handle.output_area;\n    const output = handle.output;\n\n    // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n    if ((output.output_type != \"display_data\") || (!Object.prototype.hasOwnProperty.call(output.data, EXEC_MIME_TYPE))) {\n      return\n    }\n\n    const toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n\n    if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n      toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n      // store reference to embed id on output_area\n      output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n    }\n    if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n      const bk_div = document.createElement(\"div\");\n      bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n      const script_attrs = bk_div.children[0].attributes;\n      for (let i = 0; i < script_attrs.length; i++) {\n        toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n        toinsert[toinsert.length - 1].firstChild.textContent = bk_div.children[0].textContent\n      }\n      // store reference to server id on output_area\n      output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n    }\n  }\n\n  function register_renderer(events, OutputArea) {\n\n    function append_mime(data, metadata, element) {\n      // create a DOM node to render to\n      const toinsert = this.create_output_subarea(\n        metadata,\n        CLASS_NAME,\n        EXEC_MIME_TYPE\n      );\n      this.keyboard_manager.register_events(toinsert);\n      // Render to node\n      const props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n      render(props, toinsert[toinsert.length - 1]);\n      element.append(toinsert);\n      return toinsert\n    }\n\n    /* Handle when an output is cleared or removed */\n    events.on('clear_output.CodeCell', handleClearOutput);\n    events.on('delete.Cell', handleClearOutput);\n\n    /* Handle when a new output is added */\n    events.on('output_added.OutputArea', handleAddOutput);\n\n    /**\n     * Register the mime type and append_mime function with output_area\n     */\n    OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n      /* Is output safe? */\n      safe: true,\n      /* Index of renderer in `output_area.display_order` */\n      index: 0\n    });\n  }\n\n  // register the mime type if in Jupyter Notebook environment and previously unregistered\n  if (root.Jupyter !== undefined) {\n    const events = require('base/js/events');\n    const OutputArea = require('notebook/js/outputarea').OutputArea;\n\n    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n      register_renderer(events, OutputArea);\n    }\n  }\n  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n    root._bokeh_timeout = Date.now() + 5000;\n    root._bokeh_failed_load = false;\n  }\n\n  const NB_LOAD_WARNING = {'data': {'text/html':\n     \"<div style='background-color: #fdd'>\\n\"+\n     \"<p>\\n\"+\n     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    const el = document.getElementById(\"5376\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error(url) {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (let i = 0; i < css_urls.length; i++) {\n      const url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    for (let i = 0; i < js_urls.length; i++) {\n      const url = js_urls[i];\n      const element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.async = false;\n      element.src = url;\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.4.3.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-2.4.3.min.js\"];\n  const css_urls = [];\n\n  const inline_js = [    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\nfunction(Bokeh) {\n    }\n  ];\n\n  function run_inline_js() {\n    if (root.Bokeh !== undefined || force === true) {\n          for (let i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\nif (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      const cell = $(document.getElementById(\"5376\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));",
      "application/vnd.bokehjs_load.v0+json": ""
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from bokeh.io import push_notebook, show, output_notebook\n",
    "from bokeh.plotting import figure\n",
    "output_notebook()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "id": "523cfc8e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e^{x}$"
      ],
      "text/plain": [
       "exp(x)"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp = sp.exp(var)\n",
    "exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "id": "71459d9c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x^{6}}{720} + \\frac{x^{5}}{120} + \\frac{x^{4}}{24} + \\frac{x^{3}}{6} + \\frac{x^{2}}{2} + x + 1$"
      ],
      "text/plain": [
       "x**6/720 + x**5/120 + x**4/24 + x**3/6 + x**2/2 + x + 1"
      ]
     },
     "execution_count": 139,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maclaurin_pol = ma.polynomial(p, q, var, exp)\n",
    "maclaurin_pol\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "id": "c859dc66",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\frac{x^{3}}{120} + \\frac{x^{2}}{10} + \\frac{x}{2} + 1}{- \\frac{x^{3}}{120} + \\frac{x^{2}}{10} - \\frac{x}{2} + 1}$"
      ],
      "text/plain": [
       "(x**3/120 + x**2/10 + x/2 + 1)/(-x**3/120 + x**2/10 - x/2 + 1)"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pade_33 = da.pade(p, q, var, coeffs_list)\n",
    "pade_33"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "id": "dffc6075",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "id": "03080c90",
   "metadata": {},
   "outputs": [],
   "source": [
    "x_set = np.linspace(-3,3,100)  \n",
    "exp_y = [float(exp.subs(var, i)) for i in x_set]\n",
    "maclaurin_pol_y = [ float(maclaurin_pol.subs(var, i)) for i in x_set]\n",
    "pade_33_y = [float(pade_33.subs(var, i)) for i in x_set]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "id": "916470ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div class=\"bk-root\" id=\"8bc1709d-148b-4703-8158-a690f45c30b3\" data-root-id=\"7276\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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   "source": [
    "p = figure(plot_width = 600, plot_height = 500)\n",
    "p.line(x_set, exp_y, color = 'black', legend_label= 'exp(x)')\n",
    "p.line(x_set, maclaurin_pol_y, color = 'red', legend_label= 'maclaurin(x)')\n",
    "p.line(x_set, pade_33_y, color = 'green', legend_label='[3/3](x)')\n",
    "show(p)"
   ]
  }
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